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High School Math : Intermediate Single-Variable Algebra

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Example Question #1571 : High School Math

Factor the following polynomial:

x^2-9-6xy+9y^2

Possible Answers:

(x-3y-3)(x-3y-3)

(x-3y+3)(x-3y-3)

(x-3y+3)(x+3y-3)

(x+3y+3)(x-3y-3)

(x+3y+3)(x+3y+3)

Correct answer:

(x-3y+3)(x-3y-3)

Explanation:

Begin by rearranging the terms to group together the quadratic:

x^2-9-6xy+9y^2

(x^2-6xy+9y^2)-9

Now, convert the quadratic into a square:

(x-3y)^2-9

Finally, distribute the :

(x-3y+3)(x-3y-3)

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Example Question #1572 : High School Math

Factor the following polynomial:

m^4n+m^2n-mn^4-mn^2

Possible Answers:

mn(m-n)(m^2+mn-n^2+1)

mn(m-n)(m^2+mn+n^2-1)

mn(m+n)(m^2+mn+n^2+1)

mn(m-n)(m^2-mn+n^2+1)

mn(m-n)(m^2+mn+n^2+1)

Correct answer:

mn(m-n)(m^2+mn+n^2+1)

Explanation:

Begin by extracting from the polynomial:

m^4n+m^2n-mn^4-mn^2

mn(m^3+m-n^3-n)

Now, rearrange to combine like terms:

mn(m^3-n^3+m-n)

Extract the like terms and factor the cubic:

mn[(m-n)(m^2+mn+n^2)+1(m-n)]

Simplify by combining like terms:

mn(m-n)(m^2+mn+n^2+1)

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Example Question #1573 : High School Math

Factor the following polynomial:

3x^3+9x^2+9x+3

Possible Answers:

3(x-1)^2(x+1)

3(x+1)^2(x-1)

3(x-1)^3

3(x+1)^3

Correct answer:

3(x+1)^3

Explanation:

Begin by extracting from the polynomial:

3x^3+9x^2+9x+3

3(x^3+3x^2+3x+1)

Now, rearrange to combine like terms:

3(x^3+1+3x^2+3x)

Extract the like terms and factor the cubic:

3[(x+1)(x^2-x+1)+3x(x+1)]

Simplify by combining like terms:

3[(x+1)(x^2-x+1+3x)]

3(x+1)(x^2+2x+1)

3(x+1)(x+1)(x+1)

3(x+1)^3

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Example Question #1 : Write A Polynomial Function From Its Zeros

Consider the equation $14x^{4} +Ax^{3}+Bx^{2}+Cx+6 = 0$ .

According to the Rational Zeroes Theorem, if A,B,C are all integers, then, regardless of their values, which of the following cannot be a solution to the equation?

Possible Answers:

$x = \frac{6}{7}$

$x=\frac{10}{3}$

x=3

$x=\frac{1}{2}$

$x= \frac{3}{14}$

Correct answer:

$x=\frac{10}{3}$

Explanation:

By the Rational Zeroes Theorem, any rational solution must be a factor of the constant, 6, divided by the factor of the leading coefficient, 14.

Four of the answer choices have this characteristic:

$3 = 3 \div 1$

$\frac{1}{2} = 1\div 2$

$\frac{6}{7} = 6 \div 7$

$\frac{3}{14} =3 \div14$

$\frac{10}{3}$ is in lowest terms, and 3 is not a factor of 14. It is therefore the correct answer.

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Example Question #1 : Simplifying And Expanding Quadratic Equations

Solve the following equation using the quadratic form:

x^4-9=0

Possible Answers:

$x=\pm\sqrt{4}, \pm i\sqrt{4}$

$x=\pm\sqrt{5}, \pm i\sqrt{5}$

$x=\pm\sqrt{2}, \pm i\sqrt{2}$

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

$x=\pm\sqrt{6}, \pm i\sqrt{6}$

Correct answer:

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

Explanation:

Factor the equation and solve:

x^4-9=0

(x^2-3)(x^2+3)=0

x^2 = 3

$x = \pm \sqrt{3}$

x^2 = -3

$x = \pm i\sqrt{3}$

Therefore there are four answers:

$x=\pm\sqrt{3}, \pm i\sqrt{3}$

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Example Question #2 : Simplifying And Expanding Quadratic Equations

Solve the following equation using the quadratic form:

$x-4\sqrt{x}-45=0$

Possible Answers:

x=36

x=25

x=64

x=49

x=81

Correct answer:

x=81

Explanation:

Factor the equation and solve:

$x-4\sqrt{x}-45=0$

$(\sqrt{x}-9)(\sqrt{x}+5)=0$

$\sqrt{x}=9$

x=81

$\sqrt{x}=-5$

This has no solutions.

Therefore there is only one answer:

x=81

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Example Question #1 : Using Foil

Evaluate $(2x+2y)^{2}$

Possible Answers:

$4x^{2}+8xy+4y^{2}$

$\dpi{100} x^{2}+xy+y^{2}$

$\dpi{100} 2x^{2}+4xy+2y^{2}$

$\dpi{100} 4x^{2}+4y^{2}$

$\dpi{100} 4x^{2}+4xy+4y^{2}$

Correct answer:

$4x^{2}+8xy+4y^{2}$

Explanation:

In order to evaluate $\dpi{100} (2x+2y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms.

$\dpi{100} \dpi{100} (2x+2y)^{2}=(2x+2y)(2x+2y)$

Multiply terms by way of FOIL method.

=(2x*2x)+(2x*2y)+(2y*2x)+(2y*2y)

Now multiply and simplify.

$=4x^{2}+4xy+4xy+4y^{2}$

$\rightarrow 4x^{2}+8xy+4y^{2}$

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Example Question #2 : Using Foil

Evaluate $(x-2)^{2}$

Possible Answers:

$\dpi{100} x^{2}+4$

$\dpi{100} x^{2}-4$

$x^{2}-4x+4$

$\dpi{100} 4x^{2}-4x+4$

$\dpi{100} x^{2}-2x+4$

Correct answer:

$x^{2}-4x+4$

Explanation:

In order to evaluate $\dpi{100} \dpi{100} (x-2)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

$(x-2)^{2}=(x-2)(x-2)$

Multiply terms by way of FOIL method.

=(x*x)+(x*-2)+(-2*x)+(-2*-2)

Now multiply and simplify, paying attention to signs.

$=x^{2}+(-2x)+(-2x)+4$

$\rightarrow x^{2}-4x+4$

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Example Question #3 : Using Foil

Evaluate (x+2y)*(2x-y)

Possible Answers:

$\dpi{100} 2x^{2}+4xy-2y^{2}$

$\dpi{100} 4x^{2}+xy-4y^{2}$

$\dpi{100} -2x^{2}-3xy+2y^{2}$

$2x^{2}+3xy-2y^{2}$

$\dpi{100} x^{2}+2xy-y^{2}$

Correct answer:

$2x^{2}+3xy-2y^{2}$

Explanation:

In order to evaluate (x+2y)*(2x-y) one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

Multiply terms by way of FOIL method.

=(x*2x)+(x*-y)+(2y*2x)+(2y*-y)

Now multiply and simplify, paying attention to signs.

$=2x^{2}+(-xy)+4xy+(-2y^{2})$

$\rightarrow 2x^{2}+3xy-2y^{2}$

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Example Question #4 : Using Foil

Evaluate $(x-y)^{2}$

Possible Answers:

$\dpi{100} x^{2}-y^{2}$

$\dpi{100} 2x^{2}-4xy+2y^{2}$

$x^{2}-2xy+y^{2}$

$x^{2}+y^{2}$

$\dpi{100} -x^{2}+2xy-y^{2}$

Correct answer:

$x^{2}-2xy+y^{2}$

Explanation:

In order to evaluate $\dpi{100} \dpi{100} (x-y)^{2}$ one needs to multiply the expression by itself using the laws of FOIL. In the foil method, one multiplies in the following order: first terms, outer terms, inner terms, and last terms. Be sure to pay attention to signs.

$(x-y)^{2}=(x-y)(x-y)$

Multiply terms by way of FOIL method.

=(x*x)+(x*-y)+(-y*x)+(-y*-y)

Now multiply and simplify, paying attention to signs.

$=x^{2}+(-xy)+(-xy)+(y^{2})$

$\rightarrow x^{2}-2xy+y^{2}$

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High School Math : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1571 : High School Math

Example Question #1572 : High School Math

Example Question #1573 : High School Math

Example Question #1 : Write A Polynomial Function From Its Zeros

Example Question #1 : Simplifying And Expanding Quadratic Equations

Example Question #2 : Simplifying And Expanding Quadratic Equations

Example Question #1 : Using Foil

Example Question #2 : Using Foil

Example Question #3 : Using Foil

Example Question #4 : Using Foil

All High School Math Resources

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